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1 Lnear regresson Testng for non-lnearty In analytcal chemstry, lnear regresson s commonly used n the constructon of calbraton functons requred for analytcal technques such as gas chromatography, atomc absorpton spectrometry and UV-vsble spectrometry where a lnear relatonshp s expected between the workng standard concentraton of the analyte (ndependent varable) plotted on the x-axs of the scatter plot, and the nstrument response (dependent varable) plotted on the y-axs.. The general equaton whch descrbes a ftted straght lne can be wrtten as: y = a + bx where b s the gradent of the lne and a, ts ntercept wth the y-axs. The method of least-squares lnear regresson s used to establsh the values of a and b. The best ft lne obtaned from least-squares lnear regresson s the lne whch mnmzes the sum of the squared dfferences between the observed (or expermental) and ftted values for y. The sgned dfference between an observed value (y) and a ftted value ( y ) s known as resdual. (Note: A ftted y value s obtaned by nsertng the x-value to the lnear equaton of the plot.) Hence, th resdual of squares of resdual SSR s: SSR = n = 1 ( y y ) y = y y and the total sum of It may be noted that the most common form of regresson s of y on x. Ths assumes that the x values are known exactly and the only error occurs n the measurement of y. However, there are a number of assumptons for a smple least-squares lnear regresson of y on x, such as: 1. The errors of the x-axs should be neglgble;. For estmatng confdence ntervals and drawng nferences, the error assocated wth the y-axs must be normally dstrbuted. If there s any doubt about the normalty, a few replcates of the y-values can be averaged as mean value tends to be normally dstrbuted even where ndvdual results are not; 3. The varance of the error n the y-values should be constant across the
2 range of nterest..e. the standard devaton should be constant. Smple least-squares regresson gves equal weght to all ponts; ths wll not be approprate f some ponts are much less precse than others; 4. Both the x- and y-data must be contnuous valued and not restrcted to ntegers, truncated or categorzed (for example, sample numbers, days of the week). There are a few ways to test f the least-squares regresson s truly lnear. a. Vsual examnaton of regresson data through ther resdual plots Plottng the resduals can help to dentfy problems wth poor or ncorrect curve fttng. If there s a good ft between the data and the regresson model, the resduals should be dstrbuted approxmately randomly around zero. There s no trend n the spread of resduals wth concentraton on x-axs as shown n the Fgure 1B example. However, the Fgure B llustrates a typcal plot of resduals that s obtaned when a straght lne s ftted through data that follow a non-lnear trend. It can be noted that the plot shows a certan curve trend nstead of randomly scattered around zero. In another scenaro, a resdual plot can exhbt a straght lne trend f the standard devaton of the y-values ncreases wth analyte concentraton. Such plot can be made when we have replcated results for each y-value FIG 1A: Scatter Plot of absorbance y on x concentraton Absorbance y = x R² = Concentraton mg/l
3 FIG 1B: Resdual Plots for data n FIG 1A Resdual FIG A: Scatter Plot of absorbance y on x concentraton y = x R² = FIG B: Resdual Plots for data n FIG A
4 b. F-Test for resdual standard devaton aganst repeatablty standard devaton The resdual standard devaton can be compared wth an ndependent estmate of the repeatablty of the y-values at a sngle x-value usng an F- test as shown n the followng equaton: where: F = y / x sr s s y / x = n = 1 ( y y ) n and s r s the stated repeatablty of the method n queston, whch normally can be found n ts standardzed method.. A sgnfcance testng can be carred out. The null hypothess for the test s H o : s y/x = s r and the alternatve hypothess s H 1 : s y/x > s r. The test therefore s a 1-taled test and the s y/x estmate has n- degrees of freedom where n s the number of pars of data n the regresson data set. At the 95% confdence level, the approprate F crtcal value s obtaned from tables for α = 0.05, v 1 = degrees of freedom for s y/x and v = degrees of freedom for sr. If the calculated F exceeds the crtcal value, the null hypothess s rejected,.e. H 1 s true. The nference s that the resduals are more wdely dspersed than can be accounted for by random error alone. Ths could be evdence of non-lnearty but a sgnfcant result could also occur f, for example, one or two observatons were based by other factors. Hence, studyng the scatter plot and plot of the resduals wll help to decde between the two. 3. ANOVA appled to resduals If expermental observatons are replcated at each value of x, applyng oneway ANOVA to the resduals obtaned usng the x-values as the groupng factor can warn of non-lnearty. A sgnfcant F value between group mean square ndcates that the group means devate from the lne more than would be expected from the repeatablty alone as represented by the wthn-group
5 mean square. Ths may pont to sgnfcant non-lnearty. Vsual nspecton of the resduals s tll advsable, however, because a varety of effects can cause a sgnfcant between group effect n the resduals, such as volumetrc errors or n the case of CRM, matrx effect. We shall show a worked example to llustrate the use of ANOVA appled to resduals n another paper. 4. Testng for sgnfcant hgher order terms Another practcal approach to evaluatng non-lnear data s to ft a more complex (hgher order) equaton to the data, such as a quadratc equaton wth second order, x, and determne whether the new equaton s a better representaton of the data. We shall also llustrate ths pont n future communcatons.
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